WEBVTT
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What is a function?
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Now, one definition of a
function is that a function is a
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rule that Maps 1 unique number
to another unique number. So In
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other words, If I start off with
a number and I apply my
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function, I finish up with
another unique number. So for
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example, let's suppose that my
function added three to any
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number I could start off with a
#2. I apply my function and I
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finish up with the number 5.
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Start off with the number 8 and
I apply my function and I finish
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up with the number 11.
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And if I start off with a number
X and I apply my function, I
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finish up with the number X +3
and we can show this
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mathematically by writing F of X
equals X plus three, where X is
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our inputs, which we often
called the arguments of the
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function and X +3 is our output.
Now suppose I had an argument of
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two, I could write down F of two
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equals. 2 + 3, which gives us an
output of five.
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Suppose I had an argument of
eight. I could write down F of
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eight equals 8 + 3, which gives
me an output of 11.
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And suppose I had an argument of
minus six. I could write F of
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minus 6. Equals minus 6 + 3,
which would give me minus three
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for my output.
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And if I had an argument of zed,
I could write down F of said
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equals zed plus 3.
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And likewise, if I had an
argument of X squared, I could
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write down F of X squared equals
X squared +3.
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Now it is with me pointing out
here that's a lower first sight.
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It might appear that we can
choose any value for argument.
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That's not always the case, as
we shall see later, but because
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we do have some choice on what
number we can pick the argument
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to be, this is sometimes called
the independent variable.
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And because output depends on
our choice of arguments, the
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output is sometimes called the
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dependent variable. Now let's
have a look at an example
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F of X equals 3 X
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squared. Minus 4.
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Now, as we said before, X is our
input which we call the argument
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and that is the independent
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variable. And our output which
is 3 X squared minus four is
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our dependent variable.
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Now we can choose different
values for arguments, which is
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often a good place to start when
we get function like this. So F
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of zero will give us 3 * 0
squared takeaway 4, which is 0 -
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4, which is just minus 4.
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If we start off with an argument
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of one. We get F of one
equals 3 * 1 squared takeaway 4,
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which is just three takeaway
four which gives us minus one.
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If we have an argument of two F
of two equals 3 * 2 squared
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takeaway, four switches 3 * 4,
which is 12 takeaway. Four gives
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us 8. And as I said before,
there's no reason why we can't
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include and negative arguments.
So if I put F of minus one in.
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The three times minus one
squared takeaway 4, which is 3 *
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1 gives us three takeaway. Four
gives us minus one.
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And F of minus 2.
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Would give us three times minus
2 squared takeaway 4 which is 3
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* 4 gives us 12 and take away
four will give us 8.
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So what are we going to do with
these results? Well, one thing
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we can do is put them into a
table to help us plot the graph
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of the function. So if we do a
table of X&F of X.
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The values were chosen for RX
column, which is. Our arguments
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are minus 1 - 2 zero one and
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two. So minus 2
- 1 zero 12.
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And the corresponding outputs
are 8 - 1.
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Minus 4 - 1 and
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8. And we can use this as I
said to help us plot the
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graph of the function.
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So just to copy the table down
again. So we've got handy.
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We have minus 2
- 1 zero 12.
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8 - 1 - 4
- 1 and 8.
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So we plot our graph.
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Of F of X on the vertical
axis, so output and arguments.
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On the horizontal axis.
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So we've got
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one too. Minus 1 -
2.
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And we've got.
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Minus 1 - 2 - 3 - 4.
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I'm going off.
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Up to 81234567 and eight. So
our first point is minus 2
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eight so we can put the
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appear. Our second point, minus
1 - 1 should appear down here.
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0 - 4.
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Here one and minus one.
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Yeah, but up on two and eight
which will appear over here
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and we can see we can draw a
smooth curve through these
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points, which will be the
graph of the function.
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OK, now why are we drawing a
graph of a function? Because
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this is quite useful to us
because we can now read off the
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output of a function for any
given arguments straight off the
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graph without the need to do any
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calculations. So for example, if
we look at two and arguments of
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two, we know that's going to
give us 8 before I do work that
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out. But if we looked and we
wanted to figure out.
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But the output would be when the
argument was one point 5. If we
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follow our lineup and across you
can see that that gives us a
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value between 2:00 and 3:00 for
the output, and if we substitute
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in 1.5 back into our original
expression for the function, you
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can see actually gives us an
exact value of 2.75.
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Now earlier on when I discussed
uniqueness, I said that a unique
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inputs had to give us a unique
output and by that what we mean
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is that for any given argument
we should get only one output.
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One of the benefits of having a
graph of a function is that we
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can check this using our ruler.
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If we line our ruler up
vertically and we move it left
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and right across the graph.
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We can make sure that the rule I
only have across is the graph
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wants at any point.
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And as we can see, that's
clearly the case in this
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example. And when that happens,
the graph is a valid function.
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Now, if we had the example.
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F of X.
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Equals root X.
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A good place to start is always
to substitute in some values for
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the arguments, so F of 0.
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This gives us the square root of
0, which is 0.
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Half of one's own arguments of
one will give us plus or minus
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one for the square root.
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F of two.
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Will give us plus or minus
1.4 just to one decimal place
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there. Half of 3.
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Which gives us the square root
of 3 gives us plus or minus 1.7.
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An F4. Will give us
square root of 4 which is just
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plus or minus 2.
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Now. If we try
to put in any negative
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arguments here you can see
that we're going to run
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into trouble because we
have to try and calculate
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the square root of a
negative number and we'll
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come back to this problem
in a second. But for now,
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let's plot the points that
we've got so far.
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So if we take out F of X
axis vertical again.
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And our arguments access X
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horizontal. We've
got
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1234.
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And there are vertical axis we
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have minus one. Minus 2.
00:10:06.200 --> 00:10:13.618
Plus one. Plus two points.
We've got zero and zero.
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One on plus one.
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And also 1A minus one.
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We've got two and
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positive 1.4. So round
about that and also to negative
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1.4. We've got three and
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positive 1.7. And we've
got three and negative 1.7.
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And finally we have four and +2.
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And four and negative 2.
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OK, and we've got enough points
here that we can draw a smooth
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curve through these points.
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At something it looks like.
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This.
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OK. Now.
As usual, we will apply our
00:11:11.152 --> 00:11:15.663
ruler test to make sure that the
function is valid and you can
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see straight away that when we
line up all the vertically and
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move it across for any given
positive arguments, we're
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getting two outputs. So
obviously we need to do
00:11:26.073 --> 00:11:28.155
something about this to make the
00:11:28.155 --> 00:11:32.867
function valid. One way to get
around this problem is by
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defining route X to take only
positive values or 0.
00:11:37.230 --> 00:11:38.860
This is sometimes called the
00:11:38.860 --> 00:11:42.606
positive square root. So in
effect we lose the bottom
00:11:42.606 --> 00:11:44.146
negative half of this graph.
00:11:44.750 --> 00:11:48.226
And obviously we also have the
issue of the negative arguments,
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and since we can't take the
square root of a negative
00:11:51.702 --> 00:11:55.810
number, we also have to exclude
these from the X axis. Now when
00:11:55.810 --> 00:11:58.654
we start talking about these
kind of restrictions, it's
00:11:58.654 --> 00:12:01.814
important that we use the right
kind of mathematical language.
00:12:02.420 --> 00:12:07.110
So the set of possible inputs is
what we call the domain, and the
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set of possible outputs is what
we call the range.
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So in this case, when we've got
RF of X equals the square root
00:12:16.356 --> 00:12:22.592
of X. We need to restrict our
domain to be X is more than or
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equal to 0, 'cause we only
wanted the positive values and
00:12:26.706 --> 00:12:31.568
zero. But we also notice that
now because we've got rid of the
00:12:31.568 --> 00:12:33.438
bottom half of the graph.
00:12:34.340 --> 00:12:39.228
The only part of the range which
are included are also more than
00:12:39.228 --> 00:12:44.868
or equal to 0. So range is
defined by F of X more than or
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equal to 0. So now we have a
00:12:47.876 --> 00:12:53.253
valid function. So what will do
now is just look at a couple
00:12:53.253 --> 00:12:56.683
more examples to pull together
everything that we've done so
00:12:56.683 --> 00:12:58.741
far and will start with this
00:12:58.741 --> 00:13:06.128
one. Let's look at the function
F of X equals 2 X squared
00:13:06.128 --> 00:13:08.224
minus three X +5.
00:13:08.770 --> 00:13:12.970
No, as usual, a good place to
start when you get a function is
00:13:12.970 --> 00:13:14.770
to substitute in some values for
00:13:14.770 --> 00:13:19.730
the arguments. So let's start
with that. So now arguments of 0
00:13:19.730 --> 00:13:24.110
would give us F of 0, which is 2
* 0 squared.
00:13:24.680 --> 00:13:31.480
Minus 3 * 0 +
5 which is just zero
00:13:31.480 --> 00:13:34.200
takeaway 0 + 5.
00:13:34.210 --> 00:13:38.760
So we get now pose A5 that if we
had an argument of one.
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We get 2 * 1
squared takeaway 3 * 1
00:13:46.810 --> 00:13:53.000
+ 5. Which gives us 2
* 1 here, which is 2.
00:13:53.880 --> 00:14:00.090
Take away 3 * 1 which is take
away three and plus five. So two
00:14:00.090 --> 00:14:04.230
takeaway three is minus 1 + 5
gives us 4.
00:14:05.120 --> 00:14:08.464
OK, we look at an
argument of two.
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We get 2 * 2 squared.
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Take away 3 * 2.
00:14:16.380 --> 00:14:23.911
I'm plus five which gives us 2 *
4, which is 8 and take away 3 *
00:14:23.911 --> 00:14:30.556
2 which is take away 6 and then
forget our plus five at the end.
00:14:30.556 --> 00:14:33.214
So eight takeaway six is 2.
00:14:33.810 --> 00:14:36.310
+5 gives us 7.
00:14:36.900 --> 00:14:40.399
OK. If we look at an argument of
00:14:40.399 --> 00:14:46.084
three. Half of three gives us
2 * 3 squared.
00:14:47.420 --> 00:14:49.290
Take away 3 * 3.
00:14:49.810 --> 00:14:51.300
And +5.
00:14:52.470 --> 00:14:56.026
So this is 2 * 9 here
00:14:56.026 --> 00:15:01.604
18. Take away 3 * 3 which is
take away nine and +5.
00:15:02.310 --> 00:15:07.205
So 18 takeaway 9 is 9 + 5
gives us 14.
00:15:08.120 --> 00:15:12.905
And last but not least, we can
also include a negative
00:15:12.905 --> 00:15:18.125
arguments, so we'll put negative
arguments of minus one. So F of
00:15:18.125 --> 00:15:22.040
minus one gives us two times
minus 1 squared.
00:15:22.690 --> 00:15:26.050
Take away three times minus one.
00:15:27.370 --> 00:15:30.220
And of course, our +5.
00:15:30.790 --> 00:15:34.790
So two times minus one squared
just gives us 2.
00:15:36.160 --> 00:15:39.706
Takeaway minus sorry takeaway
three times negative one which
00:15:39.706 --> 00:15:42.070
just gives us a plus 3.
00:15:42.950 --> 00:15:45.128
And then we've got a +5.
00:15:45.930 --> 00:15:49.188
2 + 3 + 5 just gives us 10.
00:15:49.790 --> 00:15:53.390
And what we can do is as before,
just put this into a table to
00:15:53.390 --> 00:15:55.310
make it nice and easy to make a
00:15:55.310 --> 00:16:01.595
graph of the function. So we put
it into a table of X.
00:16:02.160 --> 00:16:03.940
And F of X.
00:16:04.680 --> 00:16:12.078
For arguments we
had minus 10123.
00:16:12.940 --> 00:16:16.780
For Outputs, we had 10 five.
00:16:17.380 --> 00:16:20.605
4, Seven and
00:16:20.605 --> 00:16:24.380
14. OK, so let's see the graph
of this function then.
00:16:26.530 --> 00:16:27.740
You start off with our.
00:16:28.430 --> 00:16:31.446
F of X on the vertical axis as
00:16:31.446 --> 00:16:33.820
before. An argument.
00:16:34.330 --> 00:16:36.540
X on the horizontal axis.
00:16:37.680 --> 00:16:41.104
And we've got over 2 - 1, The
00:16:41.104 --> 00:16:47.986
One. 2. And
three, and on the vertical axis.
00:16:49.820 --> 00:16:52.748
Go to 15.
00:16:53.540 --> 00:16:59.616
OK, so our first point to plot
is minus one and 10 which will
00:16:59.616 --> 00:17:02.220
give us something there zero and
00:17:02.220 --> 00:17:05.810
five. There's a point here. One
00:17:05.810 --> 00:17:08.532
and four. It gives the points
00:17:08.532 --> 00:17:10.389
here. Two and Seven.
00:17:11.150 --> 00:17:15.099
Should be around here and
three and 14 which will be
00:17:15.099 --> 00:17:19.407
about here so we can see the
kind of shape that this
00:17:19.407 --> 00:17:23.715
function is starting to take
here. And we can draw in the
00:17:23.715 --> 00:17:24.074
graph.
00:17:34.700 --> 00:17:38.899
What we want is to say that
every input gives us only one
00:17:38.899 --> 00:17:42.775
single output, so we can get our
ruler again and just quickly
00:17:42.775 --> 00:17:47.297
check by going along and we can
see that as we go along. Our
00:17:47.297 --> 00:17:49.235
rule across is the curve once
00:17:49.235 --> 00:17:53.388
and once only. Which means that
this function is valid.
00:17:53.940 --> 00:17:59.044
However, an interesting point to
note is this point here. The
00:17:59.044 --> 00:18:03.684
minimum point which actually
occurs when X is North .75.
00:18:04.580 --> 00:18:11.184
So with X value of North .75
are outputs can take a minimum
00:18:11.184 --> 00:18:12.708
value of 3.875.
00:18:13.320 --> 00:18:17.254
So this means when we look at
our domain and range, we need to
00:18:17.254 --> 00:18:20.064
make no restrictions on the
domain because our function was
00:18:20.064 --> 00:18:27.040
valid. However. Our range has
a minimum of 3.875, so we write
00:18:27.040 --> 00:18:30.510
this. As F of X equals.
00:18:31.290 --> 00:18:34.797
Two X squared minus three X +5.
00:18:36.040 --> 00:18:42.982
And the range F of X has
always been more than or equal
00:18:42.982 --> 00:18:48.970
to 3.875. So for
the next example.
00:18:50.020 --> 00:18:54.376
What would happen if we had a
function F defined by?
00:18:54.960 --> 00:18:59.160
F of X equals
one over X.
00:19:00.370 --> 00:19:04.495
Well, that's always the first
stage is to substitute in some
00:19:04.495 --> 00:19:05.995
values for the arguments.
00:19:06.920 --> 00:19:09.040
So for F of one.
00:19:09.870 --> 00:19:13.687
The argument is one is 1 / 1
just gives US1.
00:19:14.980 --> 00:19:16.450
For Port F of two.
00:19:17.030 --> 00:19:18.730
We just get one half.
00:19:19.760 --> 00:19:22.358
F of three gives us 1/3.
00:19:22.910 --> 00:19:29.498
And F4 will
give us 1/4.
00:19:30.520 --> 00:19:35.030
And as before, we can also look
at some negative arguments.
00:19:35.710 --> 00:19:37.735
So if I look at F of minus one.
00:19:38.480 --> 00:19:42.528
Skip 1 divided by minus one,
which is just minus one.
00:19:43.350 --> 00:19:45.190
F of minus 2.
00:19:45.740 --> 00:19:50.396
Is 1 divided by minus two, which
just gives us minus 1/2?
00:19:51.880 --> 00:19:56.401
F of minus three. Same thing
will give us minus 1/3.
00:19:57.080 --> 00:19:59.380
And F of minus 4.
00:19:59.940 --> 00:20:02.200
Will give us minus 1/4?
00:20:02.770 --> 00:20:05.714
Now if we look at F of 0.
00:20:05.960 --> 00:20:09.896
We have 1 /
00:20:09.896 --> 00:20:13.270
0. Which is obviously a problem
00:20:13.270 --> 00:20:17.208
for us. Because of this
problem, we have to restrict
00:20:17.208 --> 00:20:21.720
our domain so that it does not
include the arguments X equals
00:20:21.720 --> 00:20:26.232
0. So let's have a look at
what the graph of this
00:20:26.232 --> 00:20:27.736
function actually looks like.
00:20:29.230 --> 00:20:33.169
So let's be 4F of X and are
vertical axis for the Outputs.
00:20:34.240 --> 00:20:38.174
An argument sax on
the horizontal axis.
00:20:39.640 --> 00:20:46.115
OK, so we've gone
over here 1234.
00:20:47.790 --> 00:20:54.698
And minus 1 - 2 -
3 - 4 over here.
00:20:54.810 --> 00:20:56.430
As we go.
00:20:57.030 --> 00:20:58.990
All the web to one down to minus
00:20:58.990 --> 00:21:05.260
one. Has it as well? So
we've got one and one.
00:21:06.390 --> 00:21:09.456
We've got 2
00:21:09.456 --> 00:21:12.510
1/2. 3
00:21:12.510 --> 00:21:16.046
1/3. 4
00:21:16.046 --> 00:21:23.556
one quarter.
We've got minus 1 - 1.
00:21:23.650 --> 00:21:27.046
Minus 2 -
00:21:27.046 --> 00:21:33.750
1/2. Minus
3 - 1/3.
00:21:33.750 --> 00:21:37.218
A minus four and minus 1/4.
00:21:37.770 --> 00:21:42.114
OK, and obviously we've excluded
0 from our domain. As we said
00:21:42.114 --> 00:21:44.286
before. So if we join these
00:21:44.286 --> 00:21:47.030
points up. And a smooth curve.
00:21:48.830 --> 00:21:53.288
Get something that
looks like this.
00:21:54.910 --> 00:21:59.618
Now, obviously we've excluded X
equals 0 from our domain, but
00:21:59.618 --> 00:22:03.898
it's also worth noticing here.
Thought there's nothing at the
00:22:03.898 --> 00:22:09.890
output of F of X equals 0, so
that also ends up being excluded
00:22:09.890 --> 00:22:11.174
from the range.
00:22:11.790 --> 00:22:16.548
So we actually end up with F of
X equals one over X.
00:22:17.450 --> 00:22:21.212
And we've got X not equal to 0
from the domain.
00:22:21.870 --> 00:22:26.690
And also. In the range F of X
never equals 0 either.
00:22:28.910 --> 00:22:30.595
But what's actually happening at
00:22:30.595 --> 00:22:35.408
this point? X equals 0 when the
arguments is zero. What is going
00:22:35.408 --> 00:22:40.000
on? Well, let's have a look and
will start off by having a look
00:22:40.000 --> 00:22:43.280
what happens as we get closer
and closer to 0.
00:22:43.820 --> 00:22:45.910
Now.
00:22:47.040 --> 00:22:52.352
If we start off with a value of
1/2 of one and remember F of X
00:22:52.352 --> 00:22:54.676
was just equal to one over X.
00:22:55.340 --> 00:23:01.085
Half of 1 just gives US1, so if
I get closer to 0 again, let's
00:23:01.085 --> 00:23:03.000
look at half of 1/2.
00:23:03.000 --> 00:23:05.628
At 1 / 1/2.
00:23:06.200 --> 00:23:11.294
This one over 1/2 which just
gives us 2.
00:23:11.300 --> 00:23:15.292
So about F of
00:23:15.292 --> 00:23:21.168
110th. It just gives us 1 / 1
tenth, which gives us 10.
00:23:21.820 --> 00:23:29.512
Half of one over 1000 will
just give us 1 / 1
00:23:29.512 --> 00:23:33.314
over 1000. Which gives
00:23:33.314 --> 00:23:40.236
us 1000. What about
one over 1,000,000, so F of
00:23:40.236 --> 00:23:42.564
one over a million?
00:23:42.570 --> 00:23:47.550
It's actually just in the same
way as before, just going to
00:23:47.550 --> 00:23:49.210
give us a million.
00:23:49.220 --> 00:23:52.068
So we can see.
00:23:52.710 --> 00:23:57.316
The US we get closer and closer
to zero from the right hand side
00:23:57.316 --> 00:23:59.619
as we saw on our graph before.
00:23:59.620 --> 00:24:03.230
We're getting closer and closer
to positive Infinity to the
00:24:03.230 --> 00:24:05.035
graph goes off to positive
00:24:05.035 --> 00:24:09.135
Infinity that. What happens when
we approach zero from the left
00:24:09.135 --> 00:24:10.965
hand side? Well, let's have a
00:24:10.965 --> 00:24:17.692
look. This is minus one, just
gives us 1 divided by minus one,
00:24:17.692 --> 00:24:19.556
which is minus one.
00:24:20.500 --> 00:24:23.988
F of minus 1/2.
00:24:23.990 --> 00:24:27.790
It's going to be just one
divided by minus 1/2.
00:24:28.330 --> 00:24:30.110
Just give his minus 2.
00:24:31.290 --> 00:24:34.401
Half of minus
00:24:34.401 --> 00:24:40.840
110th. Is 1 divided
by minus 110th?
00:24:40.910 --> 00:24:47.150
It just gives us minus 10 and we
can kind of see a pattern here.
00:24:47.150 --> 00:24:50.478
OK so F of minus one over 1000.
00:24:50.510 --> 00:24:52.760
Will actually give us minus
00:24:52.760 --> 00:24:59.384
1000. And F of
minus one over 1,000,000.
00:24:59.390 --> 00:25:05.542
Actually gives
us minus
00:25:05.542 --> 00:25:11.467
1,000,000. So you can see that
as we approach zero from the
00:25:11.467 --> 00:25:12.895
left or outputs approaches
00:25:12.895 --> 00:25:17.606
negative Infinity. And as we
approach zero from the right
00:25:17.606 --> 00:25:21.876
hand side are output approaches
positive Infinity, and these are
00:25:21.876 --> 00:25:25.292
very different things. OK, for
the last example.
00:25:25.940 --> 00:25:30.241
I just like to look at the
function F defined by.
00:25:30.250 --> 00:25:37.576
F of X equals one over
X minus two all squared.
00:25:38.310 --> 00:25:41.770
So as always with the examples
we've done, it's worthwhile
00:25:41.770 --> 00:25:45.230
started off by looking at some
different values for the
00:25:45.230 --> 00:25:51.234
arguments. So we start off with
an argument of minus two, so
00:25:51.234 --> 00:25:56.876
half of minus two gives us one
over minus 2 - 2 squared.
00:25:57.420 --> 00:26:03.803
Which gives us one over minus 4
squared, which is one over 16.
00:26:03.900 --> 00:26:07.388
F of minus one.
00:26:07.390 --> 00:26:14.628
Will give us one over minus 1
- 2 all squared, which gives us
00:26:14.628 --> 00:26:19.798
one over minus 3 squared which
is one over 9.
00:26:19.820 --> 00:26:26.450
Now, FO arguments of zero will
give us one over 0 - 2
00:26:26.450 --> 00:26:32.570
all squared, which is one over
minus 2 squared, which works out
00:26:32.570 --> 00:26:34.100
as one quarter.
00:26:34.960 --> 00:26:41.932
And F of one will give
us one over 1 - 2
00:26:41.932 --> 00:26:46.880
squared. Which is just one over
minus one squared, which gives
00:26:46.880 --> 00:26:48.008
us just one.
00:26:48.580 --> 00:26:55.990
OK. Half
of two gives us one over.
00:26:56.690 --> 00:27:01.730
2 - 2 or squared, which gives us
one over 0 which presents us
00:27:01.730 --> 00:27:06.050
with exactly the same problem we
had in the previous example when
00:27:06.050 --> 00:27:11.810
we had one over X and so we have
to exclude X equals 2 from the
00:27:11.810 --> 00:27:19.370
domain. Half of three gives us
one over 3 - 2 all squared.
00:27:20.220 --> 00:27:25.225
Which is one over 1 squared is
just gives US1 again.
00:27:25.230 --> 00:27:32.440
After four gives us one over 4
- 2 or squared, which gives us
00:27:32.440 --> 00:27:33.985
one over 4.
00:27:34.110 --> 00:27:42.080
After 5.
Gives us one over 5 - 2 or
00:27:42.080 --> 00:27:48.515
squared which gives us one over
3 squared which is 1 ninth and
00:27:48.515 --> 00:27:55.940
finally F of six gives us one
over 6 - 2 all squared which is
00:27:55.940 --> 00:28:01.385
one over 4 squared which works
out as one over 16.
00:28:01.940 --> 00:28:06.659
Now if we want to plot the graph
of this function will probably
00:28:06.659 --> 00:28:09.200
need to put this into a table
00:28:09.200 --> 00:28:15.520
first. So as usual, do our
table of X&F of X.
00:28:16.370 --> 00:28:23.104
OK, and we went from minus 2
- 1 zero all the way.
00:28:23.880 --> 00:28:26.298
Up to and arguments of sex.
00:28:27.160 --> 00:28:28.896
And the values we got for the
00:28:28.896 --> 00:28:31.932
Outputs. One over
00:28:31.932 --> 00:28:38.160
16. One 9th,
one quarter and
00:28:38.160 --> 00:28:44.292
1:01. One quarter,
1 ninth and
00:28:44.292 --> 00:28:47.358
one over 16.
00:28:48.270 --> 00:28:51.550
So we plot that onto.
00:28:52.100 --> 00:28:54.120
The graph as before.
00:28:57.060 --> 00:29:00.606
So we have arguments going along
the horizontal axis.
00:29:01.290 --> 00:29:03.600
And Outputs going along
the vertical axis.
00:29:04.720 --> 00:29:12.112
We've gone from minus
1 - 2 over
00:29:12.112 --> 00:29:15.808
there 123456 along this
00:29:15.808 --> 00:29:18.810
way. And then going off, we've
00:29:18.810 --> 00:29:24.960
gone too. One up here, so put in
a few of the marks 1/2.
00:29:25.540 --> 00:29:27.250
It's put in 1/4 that.
00:29:28.020 --> 00:29:34.728
Put in 3/4. OK, so we've got
minus two and 116th, which is
00:29:34.728 --> 00:29:37.824
going to come in down here.
00:29:38.340 --> 00:29:40.580
Minus one and one 9th.
00:29:41.380 --> 00:29:47.108
For coming over here zero
and one quarter.
00:29:47.110 --> 00:29:49.589
Over here. 1 on one.
00:29:50.930 --> 00:29:52.410
Right, the way up here?
00:29:53.200 --> 00:29:57.232
To an. Obviously this was the
divide by zero, so we couldn't
00:29:57.232 --> 00:30:00.592
do anything with that. We've
excluded, uh, from our domain.
00:30:01.690 --> 00:30:03.040
Three and one.
00:30:03.620 --> 00:30:07.280
Pay up. Four and one quarter.
00:30:08.840 --> 00:30:16.628
I'm here. Five and one,
9th and six and 116th.
00:30:16.630 --> 00:30:21.643
Because we've excluded X equals
2 from our domain.
00:30:23.370 --> 00:30:26.410
Put dotted line there,
so that's an asymptotes.
00:30:27.820 --> 00:30:29.626
And we can draw our curve.
00:30:31.970 --> 00:30:33.298
Up through the points.
00:30:33.980 --> 00:30:36.770
On this side.
00:30:38.190 --> 00:30:42.493
And we can see differently to
the other example where F of X
00:30:42.493 --> 00:30:44.479
is one over X this time.
00:30:45.090 --> 00:30:49.965
As we get approach to from both
the left and from the right,
00:30:49.965 --> 00:30:54.090
both of the outputs are heading
towards positive Infinity, so a
00:30:54.090 --> 00:30:57.840
little bit different, and also
because we've excluded X equals
00:30:57.840 --> 00:31:00.465
2 from the domain of function is
00:31:00.465 --> 00:31:05.470
now valid. But most also notice
that our range is never zero,
00:31:05.470 --> 00:31:07.315
and it's also never negative.
00:31:07.820 --> 00:31:10.826
So to write this out properly.
00:31:10.830 --> 00:31:16.394
Our function F of X equals one
over X minus two all squared.
00:31:17.710 --> 00:31:18.829
And we said.
00:31:19.340 --> 00:31:20.804
They are domain is restricted so
00:31:20.804 --> 00:31:26.506
it doesn't include two. And our
range is always more than 0.
00:31:27.640 --> 00:31:31.324
OK, so let's just recap on what
we've done in this unit.
00:31:32.310 --> 00:31:34.452
So firstly, the definition of a
00:31:34.452 --> 00:31:39.480
function. And that was that. A
function is a rule that Maps are
00:31:39.480 --> 00:31:42.880
unique number X to another
unique number F of X.
00:31:44.230 --> 00:31:48.050
Secondly, was the idea that
an argument is exactly the
00:31:48.050 --> 00:31:49.578
same as an input.
00:31:51.630 --> 00:31:55.950
Thirdly, we looked at the idea
of independent and dependent
00:31:55.950 --> 00:32:00.443
variables. And we said that
the input axe was the
00:32:00.443 --> 00:32:03.539
independent variable and the
output was the dependent
00:32:03.539 --> 00:32:03.926
variable.
00:32:05.330 --> 00:32:10.230
4th, we looked at the domain and
we said that the domain was the
00:32:10.230 --> 00:32:11.630
set of possible inputs.
00:32:12.620 --> 00:32:16.546
And finally we looked at the
range and we said that the range
00:32:16.546 --> 00:32:18.358
was the set of possible outputs.
00:32:19.610 --> 00:32:21.514
So now you know how to define a
00:32:21.514 --> 00:32:24.997
function. And how to find
the outputs of a function
00:32:24.997 --> 00:32:26.209
for a given argument?